Remark 1.5 The uniqueness of the discrete uniformization factor was proved by Thurston [1]} in terms of the rigidity of circle packing metrics with respect to the discrete curvature.
The existence of arbitrarily dense, \(\epsilon \) -regular, and uniform geodesic triangulations on any \((M, g)\)  can be deduced from the work by Colin de Verdiere [2]}. It provides a family of geodesic triangulations on any Riemannian torus whose angles of triangles are arbitrarily closed to \(\pi /3\)  and sizes tend to zero.

